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Quick Revision Reading Time: 8 min Essential Formulas

Continuity Quick Revision: Formulas, Theorems, and Graphs for JEE

Everything you need to remember about continuity - perfect for last-minute revision before the exam.

15+
Key Formulas
8
Important Theorems
6
Graph Types
100%
JEE Coverage

Why Continuity Matters in JEE

Continuity appears in every JEE Mathematics paper and forms the foundation for Limits, Differentiability, and Integration. Mastering these concepts can secure you 4-8 easy marks.

🎯 JEE Exam Pattern Insight

Continuity questions typically appear as:

  • Single correct choice (2-3 questions)
  • Multiple correct choices (1 question)
  • Numerical value type (1 question)
  • Often combined with differentiability concepts

🚀 Quick Navigation

Definition 3 Conditions Discontinuity Types Theorems Formulas Graphs Special Functions Checklist

1. Formal Definition of Continuity

Mathematical Definition

A function $f(x)$ is continuous at $x = a$ if:

$$ \lim_{x \to a} f(x) = f(a) $$

This means:

  • $f(a)$ exists (function is defined at $x = a$)
  • $\lim_{x \to a} f(x)$ exists
  • Both are equal

Continuity at a Point

A function $f(x)$ is continuous at $x = a$ if:

$$ \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a) $$

2. The 3 Conditions for Continuity

1. Existence Condition

$f(a)$ must exist and be finite

Violation: Function undefined at $x = a$

2. Limit Existence

$\lim_{x \to a} f(x)$ must exist

Violation: Left and right limits differ

3. Equality Condition

$\lim_{x \to a} f(x) = f(a)$

Violation: Limit ≠ function value

💡 Quick Check Method

For any continuity question, systematically check these 3 conditions in order. If any fails, the function is discontinuous at that point.

3. Types of Discontinuity

Removable Discontinuity

Condition: $\lim_{x \to a} f(x)$ exists but ≠ $f(a)$ or $f(a)$ undefined

Example: $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$

$$ \lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2 \quad \text{but} \quad f(1) \text{ undefined} $$

Jump Discontinuity

Condition: $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$ exist but are unequal

Example: $f(x) = \begin{cases} 1 & x \geq 0 \\ -1 & x < 0 \end{cases}$ at $x = 0$

$$ \lim_{x \to 0^-} = -1, \quad \lim_{x \to 0^+} = 1 $$

Infinite Discontinuity

Condition: At least one unilateral limit is infinite

Example: $f(x) = \frac{1}{x}$ at $x = 0$

$$ \lim_{x \to 0^-} = -\infty, \quad \lim_{x \to 0^+} = +\infty $$

Oscillatory Discontinuity

Condition: Function oscillates between values

Example: $f(x) = \sin\left(\frac{1}{x}\right)$ at $x = 0$

$$ \text{No limit exists as } x \to 0 $$

4. Important Theorems for JEE

Theorem 1: Algebra of Continuous Functions

If $f(x)$ and $g(x)$ are continuous at $x = a$, then:

$f(x) \pm g(x)$
$f(x) \cdot g(x)$
$\frac{f(x)}{g(x)}$ (if $g(a) \neq 0$)
$k \cdot f(x)$ (k constant)

Theorem 2: Continuity of Composite Functions

If $g(x)$ is continuous at $x = a$ and $f(x)$ is continuous at $g(a)$, then $f(g(x))$ is continuous at $x = a$.

$$ \lim_{x \to a} f(g(x)) = f\left(\lim_{x \to a} g(x)\right) = f(g(a)) $$

Theorem 3: Intermediate Value Theorem (IVT)

If $f(x)$ is continuous on $[a, b]$ and $k$ is between $f(a)$ and $f(b)$, then there exists $c \in (a, b)$ such that $f(c) = k$.

JEE Application: Used to prove existence of roots in equations.

Theorem 4: Continuity of Elementary Functions

All elementary functions are continuous in their domains:

  • Polynomials
  • Rational functions
  • Trigonometric functions
  • Exponential functions
  • Logarithmic functions
  • Inverse trigonometric functions

5. Essential Formulas & Limits

Standard Limits for Continuity

$\lim_{x \to 0} \frac{\sin x}{x} = 1$

$\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$

$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$

$\lim_{x \to 0} \frac{\log(1 + x)}{x} = 1$

$\lim_{x \to a} \frac{x^n - a^n}{x - a} = na^{n-1}$

Continuity Check Formulas

For piecewise functions at junction point $x = a$:

Check: $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = f(a)$

For rational functions:

Check denominator ≠ 0 in domain

For functions with mod:

Check behavior at critical points

6. Graphical Understanding

Continuous Function Graph

[Graph: Smooth unbroken curve]

No breaks, jumps, or holes

Removable Discontinuity

[Graph: Curve with a hole]

Limit exists but function undefined

Jump Discontinuity

[Graph: Sudden jump in values]

Left and right limits differ

7. Special Functions & Their Continuity

Greatest Integer Function $[x]$

Continuity: Discontinuous at all integer points

Type: Jump discontinuity

$$ \lim_{x \to n^-} [x] = n-1, \quad \lim_{x \to n^+} [x] = n, \quad [n] = n $$

Fractional Part Function $\{x\}$

Continuity: Discontinuous at all integer points

Type: Jump discontinuity

$$ \lim_{x \to n^-} \{x\} = 1, \quad \lim_{x \to n^+} \{x\} = 0, \quad \{n\} = 0 $$

Absolute Value Function $|x|$

Continuity: Continuous everywhere

Note: Not differentiable at $x = 0$

Signum Function $\text{sgn}(x)$

Continuity: Discontinuous at $x = 0$

Type: Jump discontinuity

$$ \text{sgn}(x) = \begin{cases} 1 & x > 0 \\ 0 & x = 0 \\ -1 & x < 0 \end{cases} $$

8. Last-Minute Revision Checklist

✅ Tick What You've Revised

⚠️ Common JEE Mistakes to Avoid

Conceptual Errors

  • Confusing continuity with differentiability
  • Forgetting to check all 3 conditions
  • Missing junction points in piecewise functions
  • Not considering domain restrictions

Calculation Errors

  • Incorrect left/right limit calculation
  • Wrong application of standard limits
  • Algebraic mistakes in rational functions
  • Sign errors with absolute values

🚀 Exam Day Strategy

For Continuity Questions:

  • Always check all 3 conditions systematically
  • For piecewise functions, check junction points
  • Remember special function behaviors
  • Use graphical intuition when stuck

Time Management:

  • Continuity questions: 2-4 minutes each
  • If stuck, move on and return later
  • Double-check boundary values
  • Verify with quick mental calculation

🎯 Quick Practice Test

Test your understanding with these JEE-style questions:

1. Check continuity of $f(x) = \begin{cases} \frac{\sin x}{x} & x \neq 0 \\ 1 & x = 0 \end{cases}$ at $x = 0$

2. Find points of discontinuity of $f(x) = \frac{1}{x^2 - 4}$

3. Discuss continuity of $f(x) = [x] + [-x]$ where $[.]$ is GIF

You're Ready for Continuity Questions!

This quick revision covers everything you need for JEE Continuity questions.

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